A MAXIMUM PRINCIPLE FOR A FRACTIONAL BOUNDARY VALUE PROBLEM WITH CONVECTION TERM AND APPLICATIONS

Mohammed Al-Refai,Kamal Pal

doi:10.3846/mma.2019.005

Abstract

We consider a fractional boundary value problem with Caputo-Fabrizio fractional derivative of order 1 < α < 2 We prove a maximum principle for a general linear fractional boundary value problem. The proof is based on an estimate of the fractional derivative at extreme points and under certain assumption on the boundary conditions. A prior norm estimate of solutions of the linear fractional boundary value problem and a uniqueness result of the nonlinear problem have been established. Several comparison principles are derived for the linear and nonlinear fractional problems.

Highlights

Caputo and Fabrizio have introduced a new fractional derivative with nonsingular kernel [12]. They replaced the power-law kernel by a decreasing exponential kernel
The novelty of the new derivative is that, there is no singular kernel and it has the ability to describe the material heterogeneities and the fluctuations of different scales [9, 10,14], which cannot be well described by classical local theories or by fractional models with singular kernel [12,13]
We used the result to construct a maximum principle for a linear fractional boundary value problem under the condition β

Summary

Introduction

Caputo and Fabrizio have introduced a new fractional derivative with nonsingular kernel [12]. Certain classes of Caputo-Fabrizio differential equations were transformed to differential equations with integer. We consider the following fractional boundary value problem of order 1 < α < 2,.

A maximum principle

Applications

Conclusions